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Theorem f1orn 6795
Description: A one-to-one function maps onto its range. (Contributed by NM, 13-Aug-2004.)
Assertion
Ref Expression
f1orn (𝐹:𝐴–1-1-ontoβ†’ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ Fun ◑𝐹))

Proof of Theorem f1orn
StepHypRef Expression
1 dff1o2 6790 . 2 (𝐹:𝐴–1-1-ontoβ†’ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ Fun ◑𝐹 ∧ ran 𝐹 = ran 𝐹))
2 eqid 2737 . . 3 ran 𝐹 = ran 𝐹
3 df-3an 1090 . . 3 ((𝐹 Fn 𝐴 ∧ Fun ◑𝐹 ∧ ran 𝐹 = ran 𝐹) ↔ ((𝐹 Fn 𝐴 ∧ Fun ◑𝐹) ∧ ran 𝐹 = ran 𝐹))
42, 3mpbiran2 709 . 2 ((𝐹 Fn 𝐴 ∧ Fun ◑𝐹 ∧ ran 𝐹 = ran 𝐹) ↔ (𝐹 Fn 𝐴 ∧ Fun ◑𝐹))
51, 4bitri 275 1 (𝐹:𝐴–1-1-ontoβ†’ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ Fun ◑𝐹))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542  β—‘ccnv 5633  ran crn 5635  Fun wfun 6491   Fn wfn 6492  β€“1-1-ontoβ†’wf1o 6496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-3an 1090  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-v 3448  df-in 3918  df-ss 3928  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504
This theorem is referenced by:  f1f1orn  6796  infdifsn  9594  efopnlem2  26015  cycpmcl  31968
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